DGZ and Lewis on probabilities in deterministic theories

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DGZ and Lewis on probabilities in deterministic
theories

• Barry Loewer
• Rutgers

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• What we regard as the obvious choice of
  primitive ontology- the basic kinds of entities
  that are the building blocks of everything else
  (except, of course, the wave function)should by
  now be clear Particles, described by their
  positions in space, changing with timesome of
  which, owing to the dynamical laws governing
  their evolutions, perhaps combine to form the
  familiar objects of daily experience. (1992 p10 )

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Creation Myth

• God made the world by creating
• 1. space-time
• 2. the Bohmian dynamical laws (Schrödinger and
  the guidance equation)
• 3. the initial wave function
• 4. the initial particle positions
• Every physical thing is composed of particles and
  every contingent physical fact supervenes on the
  actual and counterfactual motions of particles.

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Probabilities?

• But if this is all God does to create the world
  it is not clear that he has done enough or
  whether he has done the right thing to get
  quantum mechanics as it is used by physicists.
  Quantum mechanics is a probabilistic theory that
  makes claims about the probabilities of events
  e.g. that the probability that a certain particle
  will hit a screen in a particular region is such
  an such, that a spin measurement will result in
  such and such an outcome, and so on.
  Probabilities are an integral part of the kinds
  of phenomena typically taken to be quantum
  mechanical, e.g. the uncertainty principles and
  the violations of Bells inequalities.. These
  dont follow from the Bohmian dynamical laws
  alone.

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Bohmian probabilities

• Where do the probabilities in Bohmian mechanics
  come from? And what do they mean? In orthodox qm
  probabilities come in via the collapse postulate
  which- if understood realistically- is an
  indeterministic law that governs the evolution of
  the quantum state when measurements- whatever
  they are- are made. In GRW they also come in via
  an indeterministic law although without
  measurement figuring in the law. But in Bohmain
  mechanics the dynamical laws- Shroedingers and
  the guidance equation- are deterministic. In a
  deterministic theory it seems that there is
  only one place for probabilities to enter. as a
  distribution over the initial conditions of the
  universe.

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Initial condition probabilities?

• But what is the meaning of a probability
  distribution over the possible initial positions
  compatible with the initial wave function? There
  seem to be two answers 1 it is a subjective
  probability - perhaps one meeting certain
  rationality constraints- that represents
  ignorance of the initial positions of particles.
  2. it is an objective probability distribution
  over the initial positions of particles
  corresponding to some fact about the world..

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• DGZ argue that it makes no sense to construe the
  initial distribution as a probability. The reason
  is that there is only one universe. Evidently
  they have in mind frequency accounts of
  probability on which an event has a probability
  only relative to an ensemble of experiments or
  trials. The probability of e.g. heads on a coin
  flip is the frequency of heads on coin flips or
  perhaps the frequency in the infinite limit were
  the coin to be tossed infinitely many times.

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DGZ approach

• First off they drop talk of probability for talk
  of typicality. Even after having discussed this
  with Shelly I am not sure I understand their
  notion of typicality. But here is my take on it.
  Typical behavior is behavior that occurs almost
  all the time. But not any almost aways behavior
  is typical. For behavior to be typical laws have
  to be in some way responsible or partly
  responsible for the behavior occurring almost
  always. One can rely on typical behavior so it is
  rational to believe that behavior will be
  typical. So, for example, the Chicago cubs early
  exist from the playoffs was typical or appears
  so. There is a feeling of inevitability-at least
  among Cubs fans- that they will never win the
  world series- or if they do it will be a fluke.

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Typical Bohmian Worlds

• DGZ claim that a typical Bohmian world is one
  that behaves quantum mechanically.

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Problem

• There are as many (two to the alpha 0) initial
  conditions that lead to unQM worlds as there are
  that lead to QM worlds (world in which the QM
  frequencies are manifested).

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DGZ reply

• DGZ say instead that majority should be
  understood in terms of a measure of typicality.
  Specifically they suggest that ? squared be
  understood as measuring the typicality of
  possible initial particle positions compatible
  with the initial universal wave function. The
  idea is that the greater the value of this
  measure the more typical is the initial
  condition. The typicality of a proposition is
  then the sum of the typicality of the worlds at
  which it is true. A typical proposition has
  typicality 1 or very close to 1. DGZ prove

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DGZ

• In Bohmian mechanics, a property P is typical if
  it holds true for the overwhelming majority of
  histories Q(t) of a Bohmian universe. More
  precisely, suppose that ?t is the wave function
  of a universe governed by Bohmian mechanics a
  property P, which a solution Q(t) of the guiding
  equation for the entire universe can have or not
  have, is called typical if the set S0(P) of all
  initial configurations Q(0) leading to a history
  Q(t) with the property P has size very close to
  one, S0(P) ?0(q)2dq 1 - e , 0 e 1 , with
  size understood relative to the ?02
  distribution on the configuration space of the
  universe. For instance, think of P as the
  property that a particular sequence of
  experiments yields results that look random
  (accepted by a suitable statistical test),
  governed by the appropriate quantum distribution.
  One can show, using the law of large numbers,
  that P is a typical property.

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• With typicality measured relative to ? squared
  DGZ show that a certain property P of solutions
  Q(t) is typical. The property P is having a
  configuration such that, when subsystems of it
  merit effective wavefunctions, the actual
  subensembles have probability density of
  positions given by ?2. This pattern is
  precisely what needs to be the case for Borns
  rule to be appropriate. The claim that P is
  typical is another way of saying that a Law of
  Large Numbers result has been proven such that
  most (with respect to measure (6)) Bohmian
  histories Q(t) have property P. They have proved
  that- given a particular construal of most-
  most Bohmian universes are such that Borns rule
  works in them.

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Two questions

• 1. Why should anyone care about this account of
  most and its companion notion of typical? i.e.
  why should we believe that typical behavior will
  occur. Afterall a proposition can be typical but
  false and there are continummly many Bohmian
  worlds in which it false.
• 2. What facts about the world make it the case
  that (given an answer to the first question) ?
  squared is the correct measure of typicality.

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What makes a measure the correct typicality
measure

• 1. Nothing objective about the world. Rather it
  is subjective matter what one finds typical.
• 2. It is a matter of rationality what measures
  are typical.
• 3. The typicality measure(s) supervene on the
  dynamical laws.
• 4. The typicality measure(s) supervene on the
  dynamical laws and the initial condition (the
  initial wave function and intitial particle
  position) i.e. supervenes on the total physical
  history of the world and the laws..
• 5. The typicality measure(s) is a fundamental law
  or principle over and above the laws and total
  history.

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Neither 1, 2,5 are plausible

• I dont think either 1 or 2 are tenable for the
  same reason that it isnt tenable to construe the
  worlds initial probability distribution as
  subjective or dictated by rationality principles
  like the principle of indifference. Which
  measures are typical will ultimately determine
  what behaviors are lawful- e.g. that the
  uncertainty relations are laws- and that isnt a
  matter of subjective belief or even rationality
  alone.
• 5 doesnt have this problem but is quite
  unbelievable since it severs the connection
  between typicality and the laws and facts.

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Another grounding for typicality measures

• There is another line of thought that points to
  the claim that what is typical is a consequence
  of the dynamical laws. According to it we dont
  have to rely on ? squared to tell us what is
  typical. Many other measures every measure that
  is continuous with ? squared- delivers the same
  verdict concerning typicality of frequencies in
  infinite sequences of experiments.

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• Nevertheless there are infinitely many worlds
  satisfying the Bohmian dynamics that behave
  utterly unquanntum mechanical. It looks like more
  than the laws are required to ground the correct
  measure.

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Lewis account of laws

• . On Lewis account the laws that obtain in a
  world are specified by what he calls the Best
  Theory of that world. The Best Theory of a world
  is the true theory that Best combines simplicity,
  informativeness, (perhaps other scientific
  virtues e.g. comprehnsiveness). Lewis observes
  that simplicity and informativeness are typically
  at odds increase in informativeness may come at
  a cost in simplicity. But his idea is that our
  world may be such that there is a uniqe true
  theory that Best combines these two virtues so
  that increases in informativeness come only at a
  great cost in simplicity and increases in
  simplicity come only at a great cost in
  informativenss.

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• The idea is that probabilities assignment can be
  introduced into a theory so as to make a great
  gain in informativeness while keeping the theory
  fairly simple. For example, given a long sequence
  hhthhhtthtttththhtththhth.. a very informative
  description of the sequence will be very complex.
  A simple description will be quite
  uninformative. But it may be that the
  description sequence of outcomes of independent
  trials each with probability .5 for heads is both
  highly informative and also simple.

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Lewis applied to Bohmian Mechanics

• What God creates is the space time- perhaps a
  high dimensional space to accommodate the wave
  function- or perhaps he can get away with 31
  space time. He then distributes the particle
  positions and the values of the wave function (or
  whatever corresponds to the wave function that
  can be fit into 3-d) throughout all of space
  time. The laws then are given by the Best theory
  of this world. Of course, what laws there are
  will depend on the distribution. But it may be
  that the laws are a package of the two dynamical
  laws and the initial probability distribution ?
  squared. Note that on this account ? squared is
  not derived from the other laws- or dependent on
  a prior notion of typicality. And it is on an
  equal footing as far as lawfulness is concerned
  with the dynamical laws. This is why it can
  ground the lawfulness of the qm frequencies, the
  uncertainty relations, the impossibility of
  superluminal signaling and so on. It is not
  surprising that we have guessed that the initial
  probability distribution is the equivariant one.
  Its being equivariant makes it enormously simple.
  Any other distribution would be incredibly
  complicated wince it would change over time. Of
  course there is no guarantee that God created a
  world whose Best Theory is Bohmian Mechanics.

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Comparisons

• 1. DGZ the initial distribution or family of
  distributions measures typicality not probability
• Lewis the initial distribution is a probability
  distribution but not probability understood in
  terms of frequencies.
• 2. The DGZ account is not tied to any particular
  account of laws. But if laws are understood along
  Maudlins lines then it is not clear that the
  typical quantum mechanical frequencies should
  count as lawful since they are not consequences
  of the dynamical laws alone. They have the status
  of special science laws that depend on special
  intial conditions
• Lewis The initial probability distribution has
  the same nomological status as the dynaical laws.
• 3. DGZ prove that the frequencies obtained in
  measurement situations (and other quantum
  mechanical frequencies) are typical. This
  underlies the usual applications of Borns rule.
• Lewis can take over DGZs argument but now as an
  argument directly for the probabilities of the
  outcomes of experiments and other events. There
  is no need to take a detour through typicality
  and frequencies.